Theoretical and modelling investigation of pendant groups effect on quantum interference for single-molecule junctions

Quantum interference (QI) is one of the most important phenomena that affects the charge transport through single molecules. The effect of a constructive and destructive quantum interference on electronic, thermoelectric and spectroscopic properties of oligo(phenyleneethynylene) based-molecular junctions has been investigated using a combination of density functional theory (DFT) methods, tight binding (Hückel) modelling (TBHM) and quantum transport theory (QTT). Molecules with carbonyl, diphenyl, ethane and ethynylferrocene substituents show a destructive quantum interference (DQI), which enhances thermoelectric properties of these molecules making them promising materials for thermoelectric applications.


Theoretical Methods
All calculations in this work were carried out by the implementation of DFT in the SIESTA 23 code.It is used to obtain the optimized geometries of the structures, as shown in Figure S2.SIESTA software is an acronym derived from the Spanish Initiative for Electronic Simulations with Thousands of Atoms.The quantum transport theory (QTT) implemented in GOLLUM 24 software which is a program that computes the charge, spin and electronic contribution to the thermal transport properties of multi-terminal junctions has been utilized to calculate the electronic and thermoelectric properties of all molecular junctions in this work.All, theories and computational methods and procedures are shown in Figure S1.The optimized geometries, ground state Hamiltonian and overlap matrix elements of each structure were self-consistently obtained using the SIESTA implementation of density functional theory (DFT).SIESTA employs norm-conserving pseudopotentials to account for the core electrons and linear combinations of atomic orbitals to construct the valence states.The generalized gradient approximation (GGA) of the exchange and correlation functional is used with a double-ζ polarized (DZP) basis set, a real-space grid defined with an equivalent energy cut-off of 250 Ry.The geometry optimization for each structure is performed to the forces smaller than 20 meV/Å.The mean-field Hamiltonian obtained from the converged DFT calculation was combined with GOLLUM.The transmission coefficient T(E) for electrons of energy E (passing from the source over molecule to the drain) is calculated via the relation: (1) Γ L,R describes the level broadening due to the coupling between left (L) and right (R) electrodes and the central scattering region, Σ L,R (E) are the retarded self-energies associated with this coupling. (3) G R is the retarded Green's function, where H is the Hamiltonian and Ӽ is the overlap matrix (both of them are obtained from SIESTA).The transport properties is then calculated using the Landauer formula: Electronic Supplementary Material (ESI) for RSC Advances.This journal is © The Royal Society of Chemistry 2024 where is the conductance quantum, e is electron charge and h is the Planck's constant.
where f is the Fermi-Dirac distribution function, T is the temperature and k B = 8.6×10 -5 eV/K is Boltzmann ' s constant 78 .DFT can give inaccurate value for the Fermi energy that the calculated conductance are obtained for a range of Fermi energies 25 .The thermopower or Seebeck coefficient S is defined as the difference of electrochemical potential per unit temperature difference developing across an electrically isolated sample exposed to a temperature gradient.The Seebeck coefficients and power factors is also informative.Provided the transmission function, T(E), can be approximated by a straight line on the scale of K BT , the Seebeck coefficient is given by: (6) where L is the Lorenz number WΩK -2 .In other words, S is proportional to the negative of the slope of lnT(E), evaluated at the Fermi energy.The power factor is the ratio of the real power absorbed by the load to the apparent power flowing in the circuit.Real power is the average of the instantaneous product of voltage and current and represents the capacity of the electricity for performing work.From the Seebeck coefficient, the power factor was calculated as given in equation ( 7) where T is the temperature T = 300 K, G is the electrical conductance and S is the thermopower.
In conventional devices the maximum efficiency of either heat transfer or current generation is proportional to the dimensionless thermoelectric figure of merit.The common measure for thermoelectric efficiency is given by the figure of merit, which is given by: 25 (8) where G is the electrical conductance, S is Seebeck coefficient, is the electron thermal conductance, is the phonon    ℎ thermal conductance.[28] (9) In the expressions e = |e| is the absolute value of the electron charge, h is the Planck constant, and T = (T L + T R )/2 is the average junction temperature.The coefficients in 9 and 10 are defined as: (11) where T ph (E) is the electron transmission, and the chemical potential μ ≈ E F is approximately given by the Fermi energy E F of the Au electrodes.
The corresponding thermal conductance due to the phonons is given in linear response by: ( 12) where T ph (E) is the phonon transmission and n(E,T ) = {exp(E/ kB T ) − 1} −1 is the Bose function, characterizing the phonon reservoirs in the left and right electrodes.
Hence, an upper bound for ZT in the limit of vanishing phonon thermal transport κ ph → 0 is given by the purely electronic contribution 26 as ( 13)

𝐿
Hence, the Lorenz number is L = κ el /GT.With Z el T, and depending on above the figure of merit is presented in a slightly different form as: All molecules in a gas phase has been designed using Avogadro 29 visualizer, then the ground-state energy optimization of molecules and iso-surfaces calculations achieved using Gaussian 30 software at the B3LYP level of theory 31 with 6-31G** basis set. 32,332][33] It is a so-called hybrid functional and is usually expressed in the following form: where F X Slater refers to the Slater exchange, F X HF is the Hartree-Fock exchange, F X 88 is the Becke's exchange functional.F C LYP is the correlation functional of Lee, Yang, and Parr, and F C VWN is the correlation functional of Vosko, Wilk, and Nusair.The coefficients are a = 0.20, b = 0.72, and c = 0.81, which were adapted from another hybrid functional, B3PW91.The values of the coefficients were originally determined empirically by a linear least-squares fit to 116 experimentally determined energies.The second step involves the rotation, sorting and linking the molecules to the gold electrodes to obtain the theoretical models of molecular junctions (see Figure S3), using a set of FORTRAN algorithms.After that the molecular junctions have been optimized using SIESTA. 23The data (Hamiltonian and overlap matrix) was then fed to the Gollum 24 code, which calculating the electronic and thermoelectric properties of all molecular junctions.The theoretical models of all molecular configurations consist of optimized molecules attached two (111)-directed gold electrodes involving small 6-atom pyramidal gold leads, and each electrode constructed of eight layers of (111)-oriented bulk gold with each layer consisting of 6 × 6 atoms, and a layer spacing of 0.235 nm were employed to create the molecular junctions.These layers were then further repeated to yield infinitely long current carrying gold electrodes.From these model junctions the electronic and thermoelectric properties were calculated using the GOLLUM code.

Tight Binding Hückel Model
The neglect of tight-binding Hückel model (TBHM) of the interactions between electrons is considered a major defect, but it remains one of the widely used methods to visualize and understand the electronic properties of molecular junctions. 34ne of the drawbacks of this kind of computational methods is the produced energy levels are diminished by a few eV in comparison with the accurate values relative to a vacuum, but the energy variances are usually appropriate to compare with DFT calculations.Anyway, this way is considered a powerful tool to award reasonable and precise results that could figure out the fundamental physics and resolve the problems.
In seeking to understand the transport behaviour of molecules and the relative effect of different central parts, a minimal tight-binding (Hückel) model (TBHM) has been constructed, as shown in Figure S3.The simplest tight-binding Hamiltonian of the parent is obtained by assigning a site energy ε to each diagonal and a nearest neighbor hopping integral γ between neighbouring sites, i.e., H ii = ε and H ij = γ if i, j are nearest neighbours.A minimal model of the central-substituted "daughters" (carbonyl, diphenyl, ethane, ethynylferrocene) is then obtained, as shown in Figure S3. is the coupling element of the dihedral angle between phenyl rings.The coupling element of molecular rotation of CH3 fragments is γ 3 .γ 4 is the coupling element of twisting angle.
Figure S3 shows a minimal tight-binding (Hückel) representation of m-OPE models with different pandent groups (carbonyl, diphenyl, ethyl, ethynyl ferrocene groups).The model involves a system connected to two one-dimension electrodes on both sides by weak nearest neighbor couplings γ R and γ L .
Figure S4 shows the transmission coefficints OPE-2 model with carbonyl group as a central part of the molecule in three different model cases (I,II,III).The double bonds in carbonyl group are very reactivity.This reactive due to C=O electronegativity attributed to the oxygen atom and its two lone pairs of electrons. 35,36One pair of the oxygen lone pairs are located in 2s orbital, while the other pair are in 2p orbital, where its axis is directed perpendicular to the direction of the π orbitals, as shown in Fig. ure S5.The double bond length of a carbonyl group is about 1.2 A • and the strength is about 176-179 kcal/mol. 35The Carbonyl group properties are directly tied to its electronic structure as well as geometric positioning.It is possible to correlate the length of a carbonyl bond with its polarity; the longer the bond meaing the lower the polarity.The CH3 fragments of ethane substitute are joined by a carbon-carbon sigma bonds allowing them to rotate about these bonds.In terms of a Newman projection 37 , the free rotation around single bonds, there will be various conformations for ethane compound.Up to six unique conformations may be drawn, and each conformation is drawn by rotation of either the proximal or distal atom 60 ○ .Of these six conformations, three will be in a staggered conformation, while the other three will be in an eclipsed conformation.These six conformations can be represented in a relative energy diagram, as shown in Figure S7.Staggered conformations are the lower energy arrangements.This is because each side group is 60 ○ apart from the others, so there is no torsional strain; the side groups of the front carbon are maximally spread out from those of the back carbon, which decreases the possibility of interaction.Eclipsed conformations are higher in energy than staggered due to bond straining.In this arrangement, the side groups of the front carbon are directly in front of the side groups of the back carbon; so when looking from the side you would only see the side groups of the closer carbon.The conformational interchange process from eclipsed to staggered involves a twist around the C5 rotation axis that changes the torsion angle γ connecting any carbon atom of one ring to the corresponding carbon atom of the second ring through the two ring centroids, as shown in Figure S11. Figure S10 exhibites the transmission coefficient of OPE-5 with ethynylferrocene group as a cenral part.Hückel model visualizes the the twist angle as γ 4 .For case I, γ 4 = 0.3; case II, γ 4 = 0.1, and for case III, γ 4 = 0.5.The consent between TBHM and DFT in case II.

Figure S2 .
Figure S2.Theoretical models of all molecular junctions.

Figure S3 .
Figure S3.A minimal tight-binding (Hückel) representation of all molecules with different pandent groups.γ is the coupling elements, and ε is an onsite energy.Grey balls indicates to the onsite energy of carbon atoms (ε C ), light yellow balls refers to the onsite energy of sulfur atoms (ε S ) , dark yellow balls is the onsite energy of gold atoms (ε Au ).Red balls indicates to the onsite energy of oxygen atoms (ε O ), Orange balls refers to the onsite energy of iron atoms (ε Fe ) and PG refers to pendant groups.The coupling element between carbon-carbon single bond is γ S .γ d is the coupling element between carbon-carbon double bonds.γ t is the coupling element between carbon-carbon triple bonds.γ L and γ R are the left and right coupling elements between anchor groups and gold electrodes.The coupling element between carbon-oxygen double bonds is γ 1 .γ 2

Figure S4 .
Figure S4.Transmission coefficients of OPE-2 molecule in three different model cases (I,II,III), as a function of electrons energy.The dashed black curve is the DFT-transmission coefficient.In this context, TBHM of this work visualizes the double bond length as γ 1 , which has been changed from 1.5 A • (γ 1 = 2) for the case I to 1.7 A • (γ 1 = 1) for case II and 1.9 A • (γ 1 = 0.5) for case III.All other parameters β= 1, γ= 0.5, ε C = 0.1, ε S = 2.5, ε Au = 0.5, γ S = 0.3, γ d = 0.5, γt = 0.7, γ L = γ R = 0.2, are fixed for all cases.The changing of the polar double bonds (γ 1 ) value of this structure, manipulating the electrical dipole moment between negative and positive charges of oxygen and carbon atoms respectively, which leading to different transport paths of de Broglie waves, and consequently to DQI.An exsellent agreement with DFT is obtained for case II, as shown in FigureS4.

Figure S7 .
Figure S7.Sketch shows the energy diagram of the six unique conformations of methyl fragments of the ethane substitute.37 Figure S6 illustres T(E) as a function of electrons energy of OPE-3 model in three cases.For case I, γ 3 = 0.4; case II, γ 3 = 0.7, and for case III, γ 3 = 0.3.The perfect consistency between DFT and TBHM for case II, as shown in Figure S6.

Figure S8 .
Figure S8.Transmission coefficients of OPE-4 molecule in three different model cases (I,II,III), as a function of electrons energy.The dashed black curve is the DFT-transmission coefficient.

Figure
FigureS8shows the transmission coefficints of OPE-4 model with Y-shaped enediyne (diphenyl group) as a central part of the molecule in three different model cases (I,II,III).A diphenyl compound involves two phenyl rings linked to carbon atom by single carbo-carbon bonds.The aromaticity of diphenyl is distinguished by the π-conjugation property.37,39It is conformational enantiomorphism as shown in FigureS9, since diphenyl itself is not planar, one phenyl ring being slightly twisted or canted in relation to the other as a consequence of steric crowding.The resulting chiral conformation, having a dihedral angle of about 45 ○ , equilibrates rapidly with its enantiomer by rotation about the connecting single bond.Note that a conformation having a 90 ○ dihedral angle is achiral, as a consequence of a plane of symmetry.

Figure S10 .
Figure S10.Transmission coefficients of OPE-5 molecule in three different model cases (I,II,III), as a function of electrons energy.The dashed black curve is the DFT-transmission coefficient.The OPE-5 Model contains an ethynylferrocene group, which possess an iron metal accommodated between two rings of cyclopentadienyl (CP), as shown in FigureS3.Ferrocene is an example of a broader class of 3D metallocenes which posses high chemical and thermal stability.40It consists of two cyclopentadienyl (Cp) rings bound by a central iron atom, which may rotate about the Cp-Fe-Cp axis.In the condensed phase the ground state is found to be the staggered configuration, while in the gas phase it corresponds to the eclipsed configuration.For the isolated molecule the difference between the staggered and the eclipsed configurations is about ≈ 20 meV.40

Figure S11 .
Figure S11.Sketch shows a) Eclipsed and staggered configurations of ferrocene compound; b) the twist and bend angles (γ and δ respectively)39 .